The cooking power of a microwave on MSW is dependent on the peak voltage of the waveform, which varies with different makes/models, and the DC input voltage to the inverter. So users with different makes/models will see different results, and an individual user will see a difference depending on the battery SOC. In some cases this behavior is an advantage, as a user with a smaller battery or inverter that won't support the full cooking power might be able to run the microwave from the battery at the reduced power, then have the full cooking power available when on shore power.
With the old-school simple power supply design found in most microwaves, which uses a half wave rectifier to charge a capacitor, the peak voltage of the waveform determines how much power the magnetron produces.
